y p 2 p 1 y p Flexible System p 2 y p2 p1 y p u, y 1 -u, y 2 Component Breakdown z 1 w 1 1 y p 1 P 1 P 2 w 2 z 2

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1 ROBUSTNESS OF FLEXIBLE SYSTEMS WITH COMPONENT-LEVEL UNCERTAINTIES Peiman G. Maghami Λ NASA Langley Research Center, Hamton, VA 368 Robustness of flexible systems in the resence of model uncertainties at the comonent level is considered. Secifically, an aroach for formulating robustness of flexible systems in the resence of frequency and daming uncertainties at the comonent level is resented. The synthesis of the comonents is based on a modifications of a controls-based algorithm for comonent mode synthesis. The formulation deals first ith robustness of synthesied flexible systems. It is then extended to deal ith global (non-synthesied ) dynamic models ith comonent-level uncertainties brojecting uncertainties from comonent levels to system level. A numerical examle involving a to-dimensional simulated docking roblem is orked out to demonstrate the feasibility of the roosed aroach. Introduction Comonent mode synthesis is a ell established area in modeling of flexible systems 6. There are several advantages of modeling via comonent mode synthesis. These include () It allos much indeendence in the design and analysis of comonents, i.e., the ork can be erformed by dierent grous or organiations. A good examle is the International Sace Station hich are built by dierent comanies in several countries. () It increases the oer of existing finite-element analysis and design, by alloing it to handle excessively large roblems; (3) It allos a direct synthesis of substructure test data. This is articularly useful for very large structural systems (such as the International Sace Station) that cannot be tested as a hole. It can also be used as a art of an exerimental verification tool for substructures before deloyment as a connected structure, for examle, to rovide uncertainty models for the comonent dynamics. In alications of robust control to flexible systems, uncertainty models have been tyically assumed or defined for the combined (full-u) system. This aroach is accetable so long as these uncertainties (in terms of structures and levels) have been verified through exerimental ork. Hoever, this is not the case in many situations. A good examle is here ground testing of the assembled Λ Senior Research Engineer, Guidance and Control Branch, Senior Member AIAA. Coyright cfl by the American Institute of Aeronautics and Astronautics, Inc. No coyright is asserted in the United States under Title 7, U.S. Code. The U.S. Government has a royalty-free license to exercise all rights under the coyright claimed herein for Governmental Puroses. All other rights are reserved by the coyright oner. structure is not ossible, i.e., only comonent-based testing is feasible. In such cases, reliable uncertainty models can only be defined for the comonents. Consequently, methods are needed to rovide for robust design and analysis of flexible systems ith the uncertainty defined at the comonent levels. In this aer, robustness of flexible systems ith comonent level uncertainties are addressed. First, robustness of a synthesied system to uncertainties in the frequency and daming of its comonents is considered. The aroach used for comonent synthesis is based on a slight modification of the control-based synthesis algorithm by Maghami and Lim. 7 The synthesied lant in transformed coordinates is laced in a Linear Fractional Transformation (LFT) form for robust design and analysis. Next, a formulation is also resented through hich the uncertainties at the comonent level can be extended to system level uncertainties for design or analysis. This is quite useful in situations herein hile a global model of the assembled system is available, uncertainties are defined for the comonents only. The roosed aroach is alied to a todimensional roblem involving the docking of to comonents in sace. Substructure Synthesis Assume that to flexible systems are connected as shon in Fig.. These systems may reresent to comonents of a larger flexible body (such as sacecraft or aircraft), or they may reresent to self-suicient systems that have joined for a secific urose, such as to sacecraft docking. Folloing the aroach of Maghami and Lim, 7 the interface of7

2 comonent _x A x + B u + H + E () Flexible System y C x (6) Fig. u, y -u, y Comonent Breakdon Diagram of the To-Comonent System P K s P Fig. Block Diagram of the To-Comonent System forces and moments (beteen the to comonents) may be considered as control inut forces. A block diagram for the system is illustrated in Fig.. Here, each of the comonent lants are reresented by P and P, resectively. Associated ith each comonent model, there is uncertainty in the Linear Fractional Transformation (LFT) form, reresenting arametric and nonarametric uncertainties in the comonent models. The controller, K s, denotes the controller that synthesies the combined dynamics of to comonents in a joined configuration. The state-sace reresentation of the dynamics of the comonents is comonent L x + D u u + D (8) here A i ;i; denote the state matrices for each of the comonents. Matrices B i, H i, E i, C i, F i, and L i, i ;, reresent the influence matrices for interface inuts, non-interface inuts, uncertainty oututs, interface oututs, uncertainty inuts, and non-interface outut for each comonent, resectively. The matrices D u, D, D u, and D, reresent feedthrough matrices associated ith non-interface oututs of each comonent; x and x denote the s- tate vectors of the comonents; y and y denote the interface outut vectors, including the dislacements and velocities at the interface from each comonent; and denote the non-interface outut vectors comonents hich includes erformance oututs as ell as measurement oututs for local feedback control; and denote the uncertainty inut vectors; and reresent the non-interface inuts hich include exogenous disturbances as ell as control inuts for local feedback control; and and denote the uncertainty outut vector. The influence matrices E i and F i ;i; corresonding to comonent uncertainties ill be defined later. No, the derivation in Ref. 7 is folloed to obtain the synthesied dynamics of the combined system. Hoever, in this aer, the interface outut vector includes the interface velocity vector as ell as the dislacement vector at the interface. The motivation behind including the dislacement vectors is to avoid the suerfluous rigid-body mode that may come about from the revious formulation. Let N c denote an orthonormal basis for the right null sace of C C Λ, i.e., C C Λ Nc () F x (7) _x A x + B u + H + E () y C x () F x (3) L x + D u u + D (4) and let R c denote an orthonormal comlement to N c. Transform the combined system via an orthogonal similarity transformation, such that x x N c R c Λ fi () Then, the dynamics of the combined system in transformed coordinates becomes _ ^A + ^H + ^E () of7

3 here y L ^N, ^F () ^N + ^D (3), and F ^F N F c (4) Du N L c ^B ^A D 3 () u D ^D D Du D u ^B ^H (6) and ^A ^A Λ T A Λ N c R c Nc R ^A 3 ^A4 c A ^H ^E ^B ^H ^E ^B N c R c Λ T B B Λ T H N c R c H Λ T E N c R c E (7) (8) () () Model Uncertainty The synthesied dynamics of the combined system is illustrated in Fig. 3. Here, the lant P, reresenting the system dynamics, is given as 8 _ ; 4 ^A ^H ^E ^N ^D ^F 38 ; () The uncertainty block, include uncertainties from both comonents, i.e., () The uncertainties reresented by may include arametric and nonarametric uncertainties in each of the comonents. Parametric uncertainties include uncertainties in modal frequencies, daming ratios, and mode shaes of the comonents. The nonarametric uncertainties considered here are the unmodeled dynamics of the comonents, hich ould tyically be due to the truncated modes of the comonents. In this aer, arametric uncertainties in the comonent models in the form of frequency and daming uncertainties are considered. Fig. 3 P Block Diagram of the Synthesied System Uncertainty in Frequency and Daming If, as tyically done, a modal model of the comonents are used to describe their dynamics, then uncertainties in modal frequency and daming are embedded in the elements of the comonent state matrices A and A. Consider the first comonent, and assume that there are n distinct uncertainties associated ith the A. Then, one may rite A! A + Xn i Q i i i (3) here matrix Q i reresents the influence matrix for the i i uncertainty. No, the influence matrices E and F (see Eqs. () and (3)) may be obtained from decomosition of matrices Q i ;i ; ;n. Factor each matrix as Q i E i F i (4) here E i R n r i, Ei R r i n, and r i denotes the rank of Q i. Singular Value Decomosition (SVD) is a good ay of erforming this factoriation. Then, the influence matrices E and F may be ritten as E E E E n F 6 4 F F. F n 3 7 Λ () (6) With matrices E and F defined, the synthesied system reresented by Eq. () is in suitable form for robustness analysis or control design. Comonent Uncertainty at System Level In some alications, it may be the case that an overall model of the combined structure is available either from full-u structural analysis or system identification, i.e., no comonent synthesis is required to generate the model. Hoever, the 3of7

4 model uncertainty is defined at the comonent level, here substantially more eort is tyically sent for modeling, testing, and verification. In order to erform robustness analysis or robust control design, the uncertainty formulation has to redefined at the system level. Assume, ithout any loss of generality, that the overall model at the system level is in a real modal form. Then, the model of the system may be ritten as 8 _x s ; 4 A s H s E s N s D s F s Here, A s has the folloing form ith A s x s A s A sm ir A si ii ii ir ; 3 7 (7) (8) () here ir and ii denote the real and imaginary art of the ith eigenvalue air of the system. Here, e assume that the eigenvalues are ordered from smallest to the largest. From comonent synthesis, e kno that the first m eigenvalues of matrix ^A should closely match the eigenvalues of the A s. m is tyically chosen as the number of system modes that are ithin half the bandidth of the comonent dynamics. Let matrix χ denote a real similarity transformation through hich matrix ^A is ut in real modal form, ith eigenvalue blocks in increasing order, and let Ψ χ. Such a transformation may be obtained from the comlex eigenvectors of ^A by collecting the real and imaginararts of the eigenvectors corresonding to comlex conjugate airs of eigenvalues. Alying this similarity transformation to the system in Eq. () gives 8 _fl ; 4 Ψ ^A χ Ψ ^H Ψ ^E ^N χ ^D ^F χ 38 fl ; (3) Comaring the system in Eq. (3) ith that given by Eq. (7), and keeing in mind that the first m eigenvalues of the to systems should match, it is observed that the uncertainties at the comonent level can not be ut exactly in the same structure at the system level, mainly due to the additional Fig. 4 u, y u, y m m Comonent No. Comonent No. Comonents of the Free-Free Problem dynamics in the synthesied model that is neglected. Hoever, a reasonable aroximation may be made by choosing E s and F s as E s Ψ ^E ; F s ^F χ (3) here Ψ and χ denote artitions formed from the first m ros and m columns of Ψ and χ, resectively. With E s and F s defined, the global system is reared for robust design and analysis. It should be noted that if more than one rigid-body mode are resent in the system (e.g., sace alications), then it ould be unlikely that the a rigid-body mode in ^A ould have a exact corresonding mode in A s. This is because the eigenvectors corresonding to rigid-body modes (ith reeated eigenvalues at ero) are not unique. One ay to handle this is to relace the artition in matrices A s, H s, and N s, corresonding to the rigid-body modes, ith those of Ψ ^A χ, Ψ ^H, and ^N χ, resectively. This note also alies to reeated flexible modes, excet in many ofsuch cases unique eigenvectors may exist. Numerical Results The roosed aroach for robust design and analysis under comonent-level uncertainty models is alied to a to-comonent lanar roblem shon in Figure 4. Here, a docking scenario involving to flexible sace systems is considered. The to systems are modeled as lanar free-free Euler-Bernoulli beams. The material and geometric roerties of the system ere chosen to rovide considerable modal content in the lo-mid frequency range, to make the synthesis and control design task more challenging, and are rovided in Table. It is assumed that in the docked configuration comonent no. ould rovide attitude and osition control for the combined system. 4of7

5 Table. Geometric and structural roerties Proerty Com. Com. Whole Length Mass/Length Rigidity, EI No. of modes Control Design The control design for comonent no. is briefly described. As shon in Fig. 4, there are to control inuts available for this comonent. There is a torque heel at the third of the length (xm) and there is a thruster at the middle of the beam (x7.m) to rovide attitude and osition control. Collocated ith these actuators are attitude and osition sensors, roviding measurement for the feedback loo. Here, sensors and actuators dynamics are not considered. The control design as based on the H synthesis. The erformance requirements ere defined in terms of the eighted outut sensitivity transfer function from disturbances to the measurement sensors. The eighting function as chosen as W (s+) for each channel to (s+)(s+) rovide good disturbance rejection at lo to mid frequencies as ell as to rovide integral action to minimie steady-state errors. Uncertainty in the system model, in the form of unstructured inut multilicative, as used. The eighting function, used to scale or distribute the uncertainty, as chosen as W u (s+) to emhasie model uncertainty in the mid-high frequency range. A - (s+) order model of the beam, hich included the first three flexible modes, ere used in the control design. The controller as synthesied using the "hinfsyn" routine of μ-analysis and Synthesis Toolbox. 8 The H design converged at a fl value of.4 (i.e., all the design requirements ere met), and resulted in a 6th-order controller. Figure rovides a bode lot of the controller. This controller as later used in robustness analysis of the system in the docked configuration. Robustness Analysis Here, the to comonents ere assumed to be inadocked configuration. Each of the comonents as modeled using the first modes (including to rigid-body modes and 8 flexible modes), resulting in a 4th-order state sace model for each of the substructures. The modes used ere deemed suicient to rovide a basis for describing the first Log Magnitude Phase (degrees) Frequency (radians/sec) 3 Frequency (radians/sec) Fig. Bode Plot for the H Controller for Comonent No. modes of the combined structure. The combined structure as also modeled globally (ithout synthesis) as a -m long beam using its first modes, resulting in a 4th-order state sace model. This model as used for validation of the synthesis aroach and is referred to as the truth model. Table. Natural frequencies Mode No. True Synthesied Using the rocedure outlined in Eqs. ()-(3), the dynamics of the to comonents ere synthesied (combined). Table comares the natural frequencies of the synthesied model vs. that of of7

6 Structured Singular Value..4.3 Structured Singular Value Frequency, rad/s Frequency, rad/s Fig. 6 Robust Stability μ Plot for the Synthesied System the validation (truth) model. Comarison of the frequencies indicate a good match beteen the to models. No, both the synthesied model as ell as the truth model ere assessed against uncertainties in the frequencies and daming of the to comonents. Secifically, it as assumed that there is a five ercent uncertainty associated ith the frequency of the first mode in each comonent. Furthermore, it as assumed that there is a ercent uncertainty associated ith the daming of the first mode in each comonent. The inherent structural daming ratio used for each of the comonent ere chosen at. ercent. The daming ratio used for the truth model as based on the values obtained from the synthesied model. First, robust stability of the synthesied model as assessed. Folloing Eq. (3), the coeicients Q i for the four uncertainty terms in the state matrices of the comonents, corresonding to frequency and daming in their first mode, ere defined and factored according to Eq. (4). Note that the uncertainties ere assumed to be distinct, real scalars. Using the system defined in Eq. () (ith aroriate inut/outut matrices for the uncertainties) together ith the H controller designed for comonent no., a robust stability analysis as erformed ith aid of the "mu" routine of μ-analysis and Synthesis Toolbox. 8 A Mu lot of the closed-loo system is rovided in Fig. 6, here it is observed that the structured singular value μ is less than one for all frequencies. Therefore, the synthesied system ith the H controller is robustly stable. Next, robust stability of the truth model against the same uncertainties in the comonent models as assessed. The first task as to aroximately translate the uncertainties from the comonent level to the system level. This as erformed by trans- Fig. 7 Robust Stability μ Plot for the Truth System forming the state matrix of the synthesied system to real modal form, ith eigenvalues in increasing order on the diagonal. Then, Eq. (3) as used to determine the influence matrices at the system level, and the truth model given by Eq. (6) as used in robust stability analysis. Again, the H controller designed for comonent no. as the controller used in the robustness analysis of the closed-loo truth model. A Mu lot of the closed-loo system is rovided in Fig. 7, here it is observed that the structured singular value μ is less than one for all frequencies. Comaring Figs. 6 and 7, it is observed the Mu lots for the synthesied and truth systems match each other reasonably ell. This validates the aroximate characteriation of the uncertainty at the system level from comonent level definitions. Concluding Remarks Robustness of flexible systems in the resence of model uncertainties at the comonent level as considered. Secifically, an aroach for formulating robustness of flexible systems in the resence of frequency and daming uncertainties at the comonent level as resented. A controls-based algorithm for comonent mode synthesis as folloed and modified to avoid to suerfluous modes that may come about in the dynamics. First, robustness of synthesied systems as formulated in a Linear Fractional Transformation frameork. The formulation as then extended to deal ith global (non-synthesied ) dynamic models ith comonent-level uncertainties brojecting uncertainties from comonent levels to system level. Anumerical examle involving a todimensional simulated docking roblem as orked out to demonstrate the feasibility of the roosed aroach. The results demonstrated the eectiveness of the robustness formulation in transformed coordinates and the eicacy of the rojection aroach to translate uncertainties from their comonent-level 6of7

7 structures to system-level structures. References Hurty, W., Vibrations of Structural systems by Comonent ModeSynthesis," Journal of Engineering Mechanics Division, Proc. of ASCE, 6,. 6. Craig, R.R., J. and Bamton, M., Couling of Substructures for Dynamic Analysis," AIAA Journal, Vol. 6, No. 7, 68, Hint, R., Analytical Methods in Comonent Mode Synthesis," AIAA Journal, Vol. 3, No. 8, 7, Craig, R.R., J., Methods of Comonent Mode Synthesis," Shock and Vibration Digest, Vol., No., 77,. 3. Craig, R.R., J., Structural Dynamics Introduction to Comuter Methods, Wiley, 8. 6 Meirovitch, L. and Hale, A., On the Substructure Synthesis Method," AIAA Journal, Vol., No. 7, 8, Maghami, P. G. and Lim, K. B., Substructure Synthesis A Controls Aroach," Proc. of Guidance, Navigation, and Controls Conf., Portland, OR,. 8 Balas, G. J., Doyle, J. C., Glover, K., Packard, A., and Smith, R., μ-analysis and Synthesis Toolbox, The MathWorks, Inc.,. 7of7